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On an Effective Solution of the Optimal Stopping Problem for Random Walks

Author

Listed:
  • Alexander Novikov

    (Department of Mathematical Sciences, University of Technology Sydney)

  • Albert Shiryaev

    (Mathematical Institute, Gubkina, Moscow, Russia)

Abstract

We find a solution of the optimal stopping problem for the case when a reward function is an integer function of a random walk on an infinite time interval. It is shown that an optimal stopping time is a first crossing time through a level defined as the largest root of Appell's polynomial associated with the maximum of the random walk. It is also shown that a value function of the optimal stopping problem on the finite interval {0, 1, ? , T} converges with an exponential rate as T approaches infinity to the limit under the assumption that jumps of the random walk are exponentially bounded.

Suggested Citation

  • Alexander Novikov & Albert Shiryaev, 2004. "On an Effective Solution of the Optimal Stopping Problem for Random Walks," Research Paper Series 131, Quantitative Finance Research Centre, University of Technology, Sydney.
    • Handle: RePEc:uts:rpaper:131
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    File URL: http://www.qfrc.uts.edu.au/research/research_papers/rp131.pdf
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    References listed on IDEAS

    as
    1. Zhen Liu & Philippe Nain & Don Towsley, 1999. "Bounds for a class of stochastic recursive equations," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 49(2), pages 325-333, April.
    2. Svetlana I Boyarchenko & Sergei Z Levendorskii, 2002. "Non-Gaussian Merton-Black-Scholes Theory," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4955, December.
    3. Stadje, Wolfgang, 2000. "An iterative approximation procedure for the distribution of the maximum of a random walk," Statistics & Probability Letters, Elsevier, vol. 50(4), pages 375-381, December.
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    Cited by:

    1. Jaap H. Abbring, 2012. "Mixed Hitting‐Time Models," Econometrica, Econometric Society, vol. 80(2), pages 783-819, March.
    2. Christensen, Sören & Salminen, Paavo & Ta, Bao Quoc, 2013. "Optimal stopping of strong Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 123(3), pages 1138-1159.
    3. Boyarchenko, Svetlana & Levendorskii, Sergei, 2010. "Optimal stopping in Levy models, for non-monotone discontinuous payoffs," MPRA Paper 27999, University Library of Munich, Germany.
    4. Denis Belomestny & Pavel V. Gapeev, 2006. "An Iteration Procedure for Solving Integral Equations Related to Optimal Stopping Problems," SFB 649 Discussion Papers SFB649DP2006-043, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    5. Jaap H. Abbring, 2010. "Identification of Dynamic Discrete Choice Models," Annual Review of Economics, Annual Reviews, vol. 2(1), pages 367-394, September.
    6. Alexander Novikov & Albert Shiryaev, 2006. "On a Solution of the Optimal Stopping Problem for Processes with Independent Increments," Research Paper Series 178, Quantitative Finance Research Centre, University of Technology, Sydney.
    7. Soren Christensen, 2011. "A method for pricing American options using semi-infinite linear programming," Papers 1103.4483, arXiv.org, revised Jun 2011.

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    More about this item

    Keywords

    optimal stopping; random walk; rate of convergence; Appell polynomials;
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